Part 14

88. The Apparent Size of the Moon.--The apparent size of a body is the visual angle subtended by it; that is, the angle formed by two lines drawn from the eye to two opposite points on the outline of the body. The apparent size of a body depends upon both its magnitude and its distance.

The apparent size, or angular diameter, of the moon is about thirty-one minutes. This is ascertained by means of the wire micrometer already described (19). The instrument is adjusted so that its longitudinal wire shall pass through the centre of the moon, and its transverse wires shall be tangent to the limbs of the moon on each side, at the point where they are cut by the longitudinal wire. The micrometer screw is then turned till the wires are brought together. The number of turns of the screw needed to accomplish this will indicate the arc between the wires, or the angular diameter of the moon.

In order to be certain that the longitudinal wire shall pass through the centre of the moon, it is best to take the moon when its disc is in the form of a crescent, and to place the longitudinal wire against the points, or cusps, of the crescent, as shown in Fig. 103.

89. The Real Size of the Moon.--The real diameter of the moon is a little over one-fourth of that of the earth, or a little more than two thousand miles. The comparative sizes of the earth and moon are shown in Fig. 104.

The distance and apparent size of the moon being known, her real diameter is found by means of a triangle formed as shown in Fig. 105. C represents the centre of the moon, CB the distance of the moon from the earth, and CA the radius of the moon. BAC is a triangle, right-angled at A. The angle ABC is half the apparent diameter of the moon. With the angles A and B, and the side CB known, it is easy to find the length of AC by trigonometrical computation. Twice AC will be the diameter of the moon.

The volume of the moon is about one-fiftieth of that of the earth.

90. Apparent Size of the Moon on the Horizon and in the Zenith..--The moon is nearly four thousand miles farther from the observer when she is on the horizon than when she is in the zenith. This is evident from Fig. 106. C is the centre of the earth, M the moon on the horizon, M' the moon in the zenith, and O the point of observation. OM is the distance of the moon when she is on the horizon, and OM' the distance of the moon from the observer when she is in the zenith. CM is equal to CM', and OM is about the length of CM; but OM' is about four thousand miles shorter than CM': hence OM' is about four thousand miles shorter than OM.

Notwithstanding the moon is much nearer when at the zenith than at the horizon, it seems to us much larger at the horizon.

This is a pure illusion, as we become convinced when we measure the disc with accurate instruments, so as to make the result independent of our ordinary way of judging. When the moon is near the horizon, it seems placed beyond all the objects on the surface of the earth in that direction, and therefore farther off than at the zenith, where no intervening objects enable us to judge of its distance. In any case, an object which keeps the same apparent magnitude seems to us, through the instinctive habits of the eye, the larger in proportion as we judge it to be more distant.

91. The Apparent Size of the Moon increased by Irradiation.--In the case of the moon, the word apparent means much more than it does in the case of other celestial bodies. Indeed, its brightness causes our eyes to play us false. As is well known, the crescent of the new moon seems part of a much larger sphere than that which it has been said, time out of mind, to "hold in its arms." The bright portion of the moon as seen with our measuring instruments, as well as when seen with the naked eye, covers a larger space in the field of the telescope than it would if it were not so bright. This effect of irradiation, as it is called, must be allowed for in exact measurements of the diameter of the moon.

92. Apparent Size of the Moon in Different Parts of her Orbit.--Owing to the eccentricity of the moon's orbit, her distance from the earth varies somewhat from time to time. This variation causes a corresponding variation in her apparent size, which is illustrated in Fig. 107.

93. The Mass of the Moon.--The moon is considerably less dense than the earth, its mass being only about one-eightieth of that of the earth; that is, while it would take only about fifty moons to make the bulk of the earth, it would take about eighty to make the mass of the earth.

One method of finding the mass of the moon is to compare her effect in producing the tides with that of the sun. We first calculate what would be the moon's effect in producing the tides, were she as far off as the sun. We then form the following proportion: as the sun's effect in producing the tides is to the moon's effect at the same distance, so is the mass of the sun to the mass of the moon.

The method of finding the mass of the sun will be given farther on.

94. The Orbital Motion of the Moon.--If we watch the moon from night to night, we see that she moves eastward quite rapidly among the stars. When the new moon is first visible, it appears near the horizon in the west, just after sunset. A week later the moon will be on the meridian at the same hour, and about a week later still on the eastern horizon. The moon completes the circuit of the heavens in a period of about thirty days, moving eastward at the rate of about twelve degrees a day. This eastward motion of the moon is due to the fact that she is revolving around the earth from west to east.

95. The Aspects of the Moon.--As the moon revolves around the earth, she comes into different positions with reference to the earth and sun. These different positions of the moon are called the aspects of the moon. The four chief aspects of the moon are shown in Fig. 108. When the moon is at M, she appears in the opposite part of the heavens to the sun, and is said to be in opposition; when at M' and at M''', she appears ninety degrees away from the sun, and is said to be in quadrature; when at M'', she appears in the same part of the heavens as the sun, and is said to be in conjunction.

96. The Sidereal and Synodical Periods of the Moon.--The sidereal period of the moon is the time it takes her to pass around from a star to that star again, or the time it takes her to make a complete revolution around the earth. This is a period of about twenty-seven days and a third. It is sometimes called the sidereal month.

The synodical period of the moon is the time that it takes the moon to pass from one aspect around to the same aspect again. This is a period of about twenty-nine days and a half, and it is sometimes called the synodical month.

The reason why the synodical period is longer than the sidereal period will appear from Fig. 109. S represents the position of the sun, E that of the earth, and the small circle the orbit of the moon around the earth. The arrow in the small circle represents the direction the moon is revolving around the earth, and the arrow in the arc between E and E' indicates the direction of the earth's motion in its orbit. When the moon is at M{1}, she is in conjunction. As the moon revolves around the earth, the earth moves forward in its orbit. When the moon has come round to m{1}, so that m{3}m{1} is parallel with M{3}M{1}, she will have made a complete or sidereal revolution around the earth; but she will not be in conjunction again till she has come round to M, so as again to be between the earth and sun. That is to say, the moon must make more than a complete revolution in a synodical period.

The greater length of the synodical period is also evident from Fig. 110. T represents the earth, and L the moon. The arrows indicate the direction in which each is moving. When the earth is at T, and the moon at L, the latter is in conjunction. When the earth has reached T', and the moon L', the latter has made a sidereal revolution; but she will not be in conjunction again till the earth has reached T'', and the moon L''.

97. The Phases of the Moon.--When the new moon appears in the west, it has the form of a crescent, with its convex side towards the sun, and its horns towards the east. As the moon advances towards quadrature, the crescent grows thicker and thicker, till it becomes a half-circle at first quarter. When it passes quadrature, it begins to become convex also on the side away from the sun, or gibbous in form. As it approaches opposition, it becomes more and more nearly circular, until at opposition it is a full circle. From full moon to last quarter it is again gibbous, and at last quarter a half-circle. From last quarter to new moon it is again crescent; but the horns of the crescent are now turned towards the west. The successive phases of the moon are shown in Fig. 111.

98. Cause of the Phases of the Moon.--Take a globe, half of which is colored white and the other half black in such a way that the line which separates the white and black portions shall be a great circle which passes through the poles of the globe, and rotate the globe slowly, so as to bring the white half gradually into view. When the white part first comes into view, the line of separation between it and the black part, which we may call the terminator, appears concave, and its projection on a plane perpendicular to the line of vision is a concave line. As more and more of the white portion comes into view, the projection of the terminator becomes less and less concave. When half of the white portion comes into view, the terminator is projected as a straight line. When more than half of the white portion comes into view, the terminator begins to appear as a convex line, and this line becomes more and more convex till the whole of the white half comes into view, when the terminator becomes circular.

The moon is of itself a dark, opaque globe; but the half that is towards the sun is always bright, as shown in Fig. 112. This bright half of the moon corresponds to the white half of the globe in the preceding illustration. As the moon revolves around the earth, different portions of this illumined half are turned towards the earth. At new moon, when the moon is in conjunction, the bright half is turned entirely away from the earth, and the disc of the moon is black and invisible. Between new moon and first quarter, less than half of the illumined side is turned towards the earth, and we see this illumined portion projected as a crescent. At first quarter, just half of the illumined side is turned towards the earth, and we see this half projected as a half-circle. Between first quarter and full, more than half of the illumined side is turned towards the earth, and we see it as gibbous. At full, the whole of the illumined side is turned towards us, and we see it as a full circle. From full to new moon again, the phases occur in the reverse order.

99. The Form of the Moon's Orbit.--The orbit of the moon around the earth is an ellipse of slight eccentricity. The form of this ellipse is shown in Fig. 113. C is the centre of the ellipse, and E the position of the earth at one of its foci. The eccentricity of the ellipse is only about one-eighteenth. It is impossible for the eye to distinguish such an ellipse from a circle.

100. The Inclination of the Moon's Orbit.--The plane of the moon's orbit is inclined to the ecliptic by an angle of about five degrees. The two points where the moon's orbit cuts the ecliptic are called her nodes. The moon's nodes have a westward motion corresponding to that of the equinoxes, but much more rapid. They complete the circuit of the ecliptic in about nineteen years.

The moon's latitude ranges from 5° north to 5° south; and since, owing to the motion of her nodes, the moon is, during a period of nineteen years, 5° north and 5° south of every part of the ecliptic, her declination will range from 23-1/2° + 5° = 28-1/2° north to 23-1/2° + 5° = 28-1/2° south.

101. The Meridian Altitude of the Moon.--The meridian altitude of any body is its altitude when on the meridian. In our latitude, the meridian altitude of any point on the equinoctial is forty-nine degrees. The meridian altitude of the summer solstice is 49° + 23-1/2° = 72-1/2°, and that of the winter solstice is 49° - 23-1/2° = 25-1/2°. The greatest meridian altitude of the moon is 72-1/2° + 5° = 77-1/2°, and its least meridian altitude, 25-1/2° - 5° = 20-1/2°.

When the moon's meridian altitude is greater than the elevation of the equinoctial, it is said to run high, and when less, to run low. The full moon runs high when the sun is south of the equinoctial, and low when the sun is north of the equinoctial. This is because the full moon is always in the opposite part of the heavens to the sun.